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Vector Curvature & Acceleration
AP Calculus BC · Unit 9 Vector-Valued Functions
Learning Objectives
Analyze r(t) = ⟨x(t), y(t)⟩
Velocity v(t) is tangent to curve
Acceleration a(t) pulls inward
Calculate Curvature κ and Radius R
Formulas
v(t)
= r'(t) = ⟨x', y'⟩
a(t)
= v'(t) = ⟨x'', y''⟩
Curvature κ:
κ = |x'y'' − y'x''| / (x'² + y'²)^(3/2)
Osculating Circle:
Radius R = 1 / κ
Tags
vectors
acceleration
curvature
osculating circle
VECTOR TRAJECTORY
Parabola: ⟨t, t²/4⟩
Sine Wave: ⟨t, sin(t)⟩
Spiral: ⟨t·cos(t), t·sin(t)⟩
Astroid: ⟨cos³(t), sin³(t)⟩
Time t
0.00
▶ Animate Particle
Velocity |v| (Speed)
—
Acceleration |a|
—
Curvature κ
—
Radius R (1/κ)
—
Blue arrow
: Velocity vector v(t)
Yellow arrow
: Accel vector a(t)
Green circle
: Osculating Circle (Curvature)